This paper proposes a simple formula for the fermion determinant in lattice QCD that is equivalent to the overlap formalism but more attractive in appearance. The formula is given as $ \det\frac{1+V}{2} $, where $ V = X(X^{\dagger}X)^{-1/2} $ and $ X $ is the Wilson-Dirac lattice operator with a negative mass term. This approach avoids undesired doubling and does not require fine tuning. The formula is shown to be equivalent to the overlap, which is known for preserving chiral symmetry and exact massless quarks, but is computationally expensive. The key idea stems from recent work on the overlap in odd dimensions. For $ d = 2k + 1 $, the overlap for Dirac fermions can be written as the determinant of a finite matrix. This is derived from a more explicit formulation in $ d + 1 $ dimensions where it represents Weyl fermions. Dimensional reduction leads to a simple formula for Dirac fermions in 2k dimensions. The paper provides a direct derivation without dimensional reductions. The overlap is constructed using two many-body states, which are the ground states of two bilinear Hamiltonians. The matrices $ H^{\pm} $ are derived from a matrix $ H(m) $, and the overlap is given by $ O = |<v_{+}|v_{-}>|^2 $. An equivalent representation is also given, involving states $ t_{\pm} $. The paper shows that the overlap can be written as $ |\det\frac{1+\Gamma_{2k+1}\epsilon(H^{-})}{2}| $, where $ \Gamma_{2k+1} $ is a matrix and $ \epsilon(H) $ is a function of $ H $. The matrix $ V = \Gamma_{2k+1}\epsilon(H^{-}) $ is unitary, and the determinant of $ 1 + V $ is zero, indicating exact zeros in instanton backgrounds. The paper also discusses the implications of this formula for lattice QCD, including the potential for studying chiral symmetry and the measurement of $ f_{\pi} $ using a finite-size soft-pion theorem. It also addresses the challenge of including $ \det\frac{1+V}{2} $ in dynamical simulations and the potential for unifying different approaches to regularizing chiral gauge theories. The work is supported by the DOE under grant # DE-FG05-96ER40559.This paper proposes a simple formula for the fermion determinant in lattice QCD that is equivalent to the overlap formalism but more attractive in appearance. The formula is given as $ \det\frac{1+V}{2} $, where $ V = X(X^{\dagger}X)^{-1/2} $ and $ X $ is the Wilson-Dirac lattice operator with a negative mass term. This approach avoids undesired doubling and does not require fine tuning. The formula is shown to be equivalent to the overlap, which is known for preserving chiral symmetry and exact massless quarks, but is computationally expensive. The key idea stems from recent work on the overlap in odd dimensions. For $ d = 2k + 1 $, the overlap for Dirac fermions can be written as the determinant of a finite matrix. This is derived from a more explicit formulation in $ d + 1 $ dimensions where it represents Weyl fermions. Dimensional reduction leads to a simple formula for Dirac fermions in 2k dimensions. The paper provides a direct derivation without dimensional reductions. The overlap is constructed using two many-body states, which are the ground states of two bilinear Hamiltonians. The matrices $ H^{\pm} $ are derived from a matrix $ H(m) $, and the overlap is given by $ O = |<v_{+}|v_{-}>|^2 $. An equivalent representation is also given, involving states $ t_{\pm} $. The paper shows that the overlap can be written as $ |\det\frac{1+\Gamma_{2k+1}\epsilon(H^{-})}{2}| $, where $ \Gamma_{2k+1} $ is a matrix and $ \epsilon(H) $ is a function of $ H $. The matrix $ V = \Gamma_{2k+1}\epsilon(H^{-}) $ is unitary, and the determinant of $ 1 + V $ is zero, indicating exact zeros in instanton backgrounds. The paper also discusses the implications of this formula for lattice QCD, including the potential for studying chiral symmetry and the measurement of $ f_{\pi} $ using a finite-size soft-pion theorem. It also addresses the challenge of including $ \det\frac{1+V}{2} $ in dynamical simulations and the potential for unifying different approaches to regularizing chiral gauge theories. The work is supported by the DOE under grant # DE-FG05-96ER40559.